Would AlphaGo Zero become perfect with enough training time?
Assuming you mean a mathematically perfect player, similar to what we can achieve trivially in Tic Tac Toe, then the answer is "maybe". The underlying reinforcement learning algorithms that it uses do have some convergence guarantees, but there are some caveats:


Theories of convergence that apply to value and policy functions learned by RL assume unrealistic timescales and decays of learning parameters. If you have actual infinite time and resources then it is possible to explore all board states and learn their values accurately. But then, if you have those resources at your disposal, a brute-force search would work too. 
Using neural network approximations to true value functions can put bounds on how well value functions are learned, as they rely on generalisation, and are characterised by an error metric. The values that they calculate are in fact guaranteed to not be mathematically perfect, and in part that is by design (because you want learning from similar states to apply to new unseen states, and as part of that need to accept the compromise that most state values will be slightly incorrect). This is especially true of the fast policy network used to drive Monte Carlo Tree Search (MCTS).
Running longer MCTS during play improves performance at cost of spending more time per move decision. Given infinite resources, MCTS can play perfectly, even from very crude heuristics. 


The difference between AlphaGo Zero and a more traditional game search algorithm is to do with optimal use of available computing resources, as set by available hardware, training time and decision time when playing. It is orders of magnitude more effective to use the RL self-play approach combined with the focused MCTS as in AlphaGo Zero, than any basic search. At least for similar puzzles and games, it is more efficient than any other game playing technique that has been explored. We are likely still more orders of magnitude of effort away from a perfect Go player, and there is no reason yet to suspect that this will ever become practical, and Go become "solved".